Question

A sphere is inscribed in a cube that has a surface area of 24 m${}^2$. A second cube is then inscribed within the sphere. What is the surface area in square metres of the inner cube?

• A. 6
• B. 8
• C. 4
• D. 12
• E. None of these

Answer 1

The correct option is B

8

Given that surface area of the original cube = 24 units.
$\therefore$ Length of each side of the cube =$\sqrt{\frac{24}{6}}$= 2 units.
The radius of the biggest possible sphere that can be kept inside the cube of each side 2 units.
= $\frac{2}{2}$ = 1 unit.
Now, the diagonal of the second cube inscribed in the sphere = Diameter of sphere.
$\Rightarrow$$\sqrt{3}$ (length of each side of second cube) = 2 units.
$\Rightarrow$Length of each side of second cube =$\frac{2}{\sqrt{3}}$ unit
$\therefore$ surface area of the inner cube = 6 x${\left(\frac{2}{\sqrt{3}}\right)}^2$
= $6\times \frac{4}{3}=$ 8 units